TA 

571 
LY 


IC-NRLF 


TABLE 


LOVELL 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

Class 


I 


THE  PLANE  TABLE 


AND 


ITS  USE  IN  SURVEYING 


BY 


W.  H.  LOVELL 

Topographer  U.  S.  Geological  Survey 


NEW    YORK 

McGRAW  PUBLISHING  COMPANY 
239  West  39th  Street 
1908 


GENERAL 


Copyrighted  1908 

by  the 

McGRAW  PUBLISHING  COMPANY 
NEW  YORK 


CONTENTS. 


Introduction 1 

Forms  of  Plane  Tables 3 

Adjustment  of  the  Alidade 5 

Plane  Table  Triangulation 8 

The  Three-Point  Problem 12 

The  Two-Point  Problem 26 

Centering  the  Plane  Table  over  the  Station.*.  31 

Vertical  Angulation 32 

Signals 37 

Land  Surveys 39 

Plane-Table  Traverse 42 

Projections 43 

Conclusion 45 

Index..  49 


176739 


SYMBOLS 

Triangulation  point . 

/•\     Plane-table  station. 

fi      Church. 

JE 

jj      School  house. 

U  Factory. 

Q  House  or  shop. 

E3  Barn  or  shed. 

/}  Monument. 

f  Windmill. 

T  Signal. 

|T]  Cemetery. 

Fence  Corner, 

f^  Fir  Tree. 

£j3  Deciduous  Tree. 

\  Dead  Tree. 

O  Hill  tops  and  locations  generally. 


IV 


UNIVERSITY   ) 

OF  / 


THE  PLANE  TABLE 

AND 

Its  Use  In  Surveying 

INTRODUCTION. 

The  plane  table,  one  of  the  oldest  of  surveying 
instruments,  is,  in  its  simplest  form,  merely  a 
board  for  holding  the  paper  or  other  material  upon 
which  a  map  is  drawn  with  the  aid  of  a  rule  or 
straight  edge. 

Although  the  plane  table  has  been  known  and 
used  in  Europe  to  a  greater  or  less  extent  for  more 
than  three  centuries,  at  the  present  day,  in  spite 
of  its  obvious  advantages  as  regards  speed,  economy 
convenience  and  adaptability  for  surveying  pur- 
poses, its  use  is  limited  mainly  to  government  sur- 
veys in  the  mapping  of  large  areas  of  country. 

Although  a  useful  and  serviceable  instrument 
for  railroad  and  land  surveyors,  it  has  never  come 
into  general  use  by  them  in  the  United  States. 

Why  this  is  so,  is  hard  to  explain,  unless  per- 
haps because  of  lack  of  knowledge  of  the  instru-' 
ment  and  its  methods,  as  little  has  been  written 
on  the  subject.  Of  late  years  it  is  gradually  be- 
coming better  known.  All  recent  text-books  on 
surveying  give  some  space  to  a  description  of  its 
use  and  some  instruction  is  given  on  the  subject 
in  many  of  the  colleges  and  technical  schools 
throughout  the  United  States. 

1 


2  PLANE  TABLE. 

The  plane  table  is  used  extensively  in  European 
countries  as  Gerrna.  y  Austria,  and  Italy,  etc.  on 
government  surveys  ad  ir>  he  topographical  sur- 
vey of  India  by  the  Jriti  .  government. 

In  the  United  States  ^  has  been  used  for  many 
years  by  both  the  Coast  and  Geological  Surveys, 
and  with  the  increasing  demand  for  the  mapping 
of  parks,  preserves,  municipalities  and  various 
tracts  of  land  for  public  or  private  purposes  there 
is  reason  to  believe  it  will  gradually  come  more 
and  more  into  use  as  surveyors  learn  to  appreciate 
the  advantages  of  the  instrument. 


FORMS  OF  PLANE  'fABLES. 

A  plans- table  outfit'  msis;  "  of  a  drawing-board, 
mounted  on  a  tripod,  t6%rhicn  is  attached  the  map 
or  sheet,  upon  which  the' work  is  to  be  plotted; 
a  rule  or  straight  edge,  called  an  alidade,  either 
with  or  without  a  telescope,  used  for  sighting  and 
drawing  lines  to  objects  it  is  desired  to  locate;  a 
spirit-level;  and  sometimes,  though  not  essential, 
a  declinatoire  or  compass,  to  aid  in  approximate 
orientation  of  the  table. 

An  umbrella  mounted  on  a  long  handle  with  a 
sharp  iron  point  to  admit  of  its  being  firmly 
driven  into  the  ground  is  often  used  to  protect  the 
plane  table  from  the  glare  of  the  sun. 

Numerous  kinds  of  plane  tables  have  been  in- 
vented and  used  at  different  times,  most  of  the 
variations  being  in  the  tripod  head  or  movement. 

This  in  most  cases  has  been  supplied  with  leveling 
screws,  but  in  the  Johnson  and  Gurley  tables  of 
recent  years  the  screws  have  been  dispensed  with 
and  a  universal  joint  motion  used  for  leveling  the 
table. 

The  plane  table  movement  or  tripod  head  is 
made  of  brass  and  in  the  U.S.  Coast  Survey  pattern 
three  leveling  screws  are  used,  the  horizontal  mo- 
tion is  effected  by  two  plates  sliding  upon  each 
other,  and  a  tangent  screw  is  attached  for  slow 
movement  in  azimuth. 

The  Johnson  plane  table  used  by  the  U.  S. 
3 


4  PLANE  TABLE. 

Geological  Survey  is  much  lighter,  although  suffi- 
ciently firm  and  rigid  for  all  practical  purposes. 

In  this  movement  there  are  no  leveling  screws, 
both  vertical  and  horizontal  movements  of  the 
table  being  effected  by  a  ball  and  socket  joint,  and 
the  tangent  screw  is  dispensed  with  as  the  table 
can  be  moved  in  azimuth  by  hand  with  all  suffi- 
cient accuracy.  For  triangulation  the  size  of  the 
board  commonly  used  is  24  by  36  inches. 

Folding  tables  for  transportation  on  horse-back 
are  sometimes  used  in  rough  country,  and  a  sketch 
board  made  to  serve  as  a  plane-table  in  recon- 
naissance surveys  for  military  purposes. 

Of  alidades  there  have  been  many  patterns. 
The  earliest  type,  a  flat  straight  edge  with  raised 
sights,  called  a  sight  alidade,  is  still  used  in  running 
lines  of  traverse  and  also  to  a  limited  extent  in 
plane  table  triangulation. 

The  form  of  alidade  used  by  the  U.  S.  Geological 
Survey  consists  of  a  brass  or  steel  rule  or  straight 
edge,  18  or  24  inches  long,  upon  which  the  telescope 
is  mounted  on  a  standard  four  inches  high,  the 
magnifying  power  of  the  lens  being  about  20 
diameters. 

Telescopic  alidades  are  provided  with  a  striding 
level  and  a  graduated  vertical  arc  by  means  of 
which  the  altitudes  of  all  located  points  may  be 
determined,  and  in  addition  to  the  cross-hairs, 
two  more  horizontal  hairs,  called  stadia-hairs,  are 
sometimes  added  for  use  in  stadia  work. 


ADJUSTMENTS  OF  THE  ALIDADE. 

Parallax:  Move  the  eye-piece  until  the  cross- 
hairs are  distinct  and  then  sight  some  distant 
object.  If  there  is  no  change  in  position  of  cross- 
hairs relative  to  the  object  when  position  of  eye  is 
altered,  there  is  no  parallax. 

If  position  of  cross-hairs  varies,  change  focus 
of  the  eye-piece  until  both  cross-hairs  and  object 
can  be  seen  with  distinctness  and  without  altera- 
tion of  position,  or  parallax. 

Collimation:  Place  intersection  of  vertical  and 
horizontal  lines  (cross-hairs)  of  telescope  on  some 
small  but  distinct  object  as  for  instance  a  nail- 
head.  Revolve  the  telescope  180  degrees.  If  the 
intersection  still  covers  the  object  sighted,  the 
adjustment  is  correct.  If  not,  correct  one-half 
of  the  difference  by  moving  the  diaphragm  by  the 
adjusting  screws,  again  sight  the  object,  reverse 
the  telescope,  and,  if  there  is  still  an  error  of 
collimation,  correct  one-half  by  the  adjusting 
screws  as  before  and  repeat  until  adjustment  is 
perfect. 

Striding  level:  Place  level  on  alidade  and 
bring  bubble  to  center  by  means  of  tangent-screw 
of  the  alidade.  Reverse  the  level  and  if  bubble 
remains  in  center  the  adjustment  is  perfect.  If 
not,  correct  one-half  by  tangent-screw  and  the 
other  half  by  adjusting  screw  attached  to  the 
level.  These  adjustments  should  be  made  at  every 

station. 

5 


6  PLANE  TABLE. 

Telescopic  alidades  sometimes  have  one  or  more 
levels  attached  to  the  rule  or  instead  a  circular 
level  may  be  carried  by  the  plane-tabler. 

To  adjust  the  attached  levels,  place  the  alidade 
in  the  center  of  the  table  drawing  lines  to  indicate 
its  position  and  bring  the  level  bubble  to  the  center 
by  leveling  the  table.  Reverse  the  alidade  180 
degrees  and  if  the  bubble  does  not  remain  in  the 
center,  correct  one-half  the  difference  by  adjusting 
the  level  screws  and  one-half  by  leveling  the  table. 

These  levels,  however,  are  not  essential  as  the 
striding  level  can  be  used  instead  for  leveling  the 
table,  as  well  as  for  taking  vertical  angles. 

The  alidade  rule  should  be  an  exact  straight 
edge.  To  test  its  fiducial  edge  draw  a  line  along 
it,  reverse  the  alidade  placing  it  exactly  on  the 
line  and  draw  *  another  line.  If  the  two  lines 
coincide  the  edge  is  true.  To  ascertain  if  the  sides 
of  the  alidade  rule  are  parallel  they  should  be  tested 
by  sighting  to  the  same  point.  If  they  are  not 
parallel,  care  should  be  taken  to  draw  lines  on  one 
side  only. 

Telescopic  alidades  are  of"  various  magnifying 
powers,  say  from  10  to  24  diameters. 

High  power  is  a  disadvantage  in  hazy  weather, 
as  the  particles  of  vapor  or  dust  in  the  air  are  mag- 
nified correspondingly. 

It  might  be  supposed  that  high  magnifying 
power  would  be  an  advantage,  but  this  is  not  so, 
beyond  a  certain  limit. 

Only  a   given   amount   of  light   can   enter   the 


THE  ALIDADE.  7 

object-glass  and  the  greater  the  magnifying  power 
the  less  clear  will  be  the  image. 

Two  eye-pieces,  for  high  and  low  power,  the 
former  for  clear  and  the  latter  for  hazy  weather 
would  be  a  convenience. 

The  magnifying  power  of  a  'telescope  is  the 
ratio  of  the  focal  length  of  the  eye-piece  to  that 
of  the  object  glass. 

To  ascertain  the  magnifying  power,  point  the 
telescope  towards  the  bright  sky  holding  a  slip  of 
thin  .white*  paper  before  the  eye-piece  at  the 
distance  at  wThich  the  image  of  the  object-glass 
which  is  projected  on  it,  is  well  defined. 

Divide  the  diameter  of  the  object  glass  by  the 
diameter  of  the  image. 

For  instance,  if  diameter  of  object  glass  is  two 
inches  and  the  image  0.1  inch  the  magnifying  power 
would  be  2  •*-  0.1  =  20. 


PLANE-TABLE  TRIANGULATION. 

The  plane  table  is  primarily  an  instrument  for 
graphic  triangulation. 

Instead  of  reading  and  recording  angles,  and 
afterwards  computing  the  resulting  triangles,  as 
with  a  transit  or  theodolite,  the  triangles  are 
drawn  directly  upon  the  map,  or  projection,  on 
the  plane-table  board,  thus  avoiding  possible 
errors  of  record,  adjustment  and  plotting.  By 
this  method  much  time  is  saved,  the  plane  table 
being  the  most  rapid  and  economical  instrument 
for  the  purpose  for  which  it  is  used,  that  has  yet 
been  devised.  It  has  the  additional  advantage 
that  any  point  can  usually  be  tested  from  different 
stations  in  the  field  at  time  of  survey  and  any 
error  in  its  position  detected,  all  plane-table  loca- 
tions being  exact  to  scale  employed  for  the  map. 
It  is  especially  adapted  for  use  in  country  of  con- 
siderable relief,  or  open  rolling  country,  in  fact, 
in  all  regions  except  heavily-wooded  areas  and 
plains  and  plateaus  of  low  relief. 

Although  originally  intended  for  triangulation, 
the  plane  table  has  other  uses  and  properly 
equipped  becomes  well-nigh  a  universal  surveying 
instrument,  as,  in  addition  to  triangulation,  it  is 
used  for  traverse,  topographic  sketching  and  verti- 
cal angulation,  in  short,  all  the  processes  required 
for  making  a  complete  map. 

In  beginning  plane-table  triangulation  a  section 
8 


TRIANGULATION.  9 

of  country  is  usually  selected  in  which  primary 
triangulation  stations  have  already  been  estab- 
lished. 

The  scale  and  area  to  be  surveyed  are  decided 
upon,  a  spherical  projection  is  made,  the  poly  conic 
being  used  on  nearly  all  U.  S.  Government  maps, 
and  upon  this  projection  the  primary  triangula- 
tion points  are  plotted  in  their  relative  position  in 
latitude  and  longitude. 

These,  if  possible,  should  be  at  least  three  in 
number.  A  primary  triangulation  point  is  then 
occupied  and  the  table  leveled,  either  by  the  strid- 
ing level  attached  to  the  alidade  or  a  round  level 
which  can  be  permanently  affixed  to  the  alidade 
rule  or  carried  in  the  pocket. 

Place  this  level  in  center  of  table  and  bring  the 
table  to  a  level  by  the  leveling  screws,  if  any,  or 
by  adjusting  the  legs  which  can  be  easily  done  after 
a  little  practice.  Unless  the  table  is  of  more  than 
usual  evenness  and  solidity,  it  cannot  be  brought 
to  an  exact  level  anywhere  away  from  the  center, 
but,  as  is  explained  later,  this  is  not  necessary. 
After  adjusting  the  alidade  for  parallax  and  colli- 
mation,  the  table  is  oriented  by  sighting  some 
other  triangulation  point,  that  is,  the  projections 
of  these  points  on  the  sheet  or  map  upon  which  the 
triangulation  is  to  be  done,  are  brought  into  the 
same  relative  positions  as  the  points  themselves 
on  the  ground.  Lines  are  then  drawn  to  all  ob- 
jects in  the  landscape,  such  as  signals,  church 
spires,  cupolas,  chimneys,  flag-poles,  prominent 


10  PLANE  TABLE. 

trees,  hill- tops,  spurs,  etc.,  that  may  be  of  use 
in  obtaining  proper  control  for  the  resulting  map. 
Tangents  should  also  be  drawn  to  all  railroads,  high- 
ways, shore  lines  and  other  objects  in  the  landscape 
that  it  may  not  be  possible  to  locate  by  intersection. 

Lines  drawn  to  signals  and  other  points  avail- 
able for  plane-table  stations  are  called  foresights. 

The  points  to  which  foresights  have  been  drawn 
are  occupied  and  the  table  again  oriented  by 
placing  the  alidade  on  the  foresight  and  sighting 
back  to  station  from  which  foresight  wras  taken. 

Another  located  point  is  then  observed  and  the 
position  of  point  occupied  determined  by  resection. 

A  third  located  point  is  then  sighted  and,  if  the 
line  from  this  passes  through  intersection  of  the 
lines  from  the  other  two  points,  the  table  is  in 
position  and  location  of  station  correctly  deter- 
mined. 

All  objects  to  which  lines  were  drawn  from  the 
first  station  that  can  be  identified,  are  sighted  and 
located  by  intersection  and  lines  drawn  to  other 
objects  that  come  into  view  as  well,  and  in  this 
way  by  the  use  of  foresights  an  entire  sheet  may 
be  plane-tabled,  but  it  usually  happens  sooner 
or  later  that  resort  must  be  had  to  the  three-point 
method  of  location. 

1.  It  may  occur  that  none  of  the  triangulation 
points  are  accessible  and  in  that  case  it  will  be 
necessary    to    begin    plane-tabling    with    a    three- 
point  station. 

2.  It  may  be  found  on  occupying  a  point  to 


TRIANGULAT1ON.  11 

which  a  foresight  has  been  drawn  that  lines  from 
other  located  points  will  not  intersect,  but  form  a 
triangle, — the  triangle  of  error — caused  by  an  error 
in  the  foresight  or  in  identification  of  point  to  which 
foresight  was  drawn.  In  that  case  the  table  is 
out  of  position  and  must  be  brought  into  position 
by  the  three-point  method. 

3.  In  a  country  of  great  relief  where  the  triangu- 
lation  points  are  at  a  considerable  elevation  above 
the  valleys  it  will  often  be  found  very  difficult  to 
recognize  objects  to  which  lines  have  been  drawn. 
In  that  case  it  -is  often  better  and  more  convenient 
to  make  three-point  stations  in  the  valleys  or  on 
the  lower  hills,  the  commanding  positions  of  the 
triangulation  points  making  them  easily  discernible 
with    the    additional    advantage    that    objects    to 
which  lines  have  been  drawn  can. more  easily  be 
identified. 

4.  Much  time  may  be  saved  by  the  use  of  three- 
point  stations  as  fewer  signals  are  necessary,  and 
more  stations  can  be  made  and  more  points  lo- 
cated in  a  given  time. 


THE  THREE-POINT  PROBLEM. 

The  three-point  problem  presents  no  difficulties, 
and  no  one  can  be  considered  thoroughly  compe- 
tent for  plane-table  work  who  does  not  understand 
it. 

The  rules  to  be  remembered  are  few  and  simple. 
In  making  a  three-point  station  the  plane  table 
is  set  up  in  such  position  that  at  least  three  pre- 
viously located  points  are  visible  and  in  such  rela- 
tion to  each  other  and  the  point  to  be  determined 
that  lines  drawn  from  them  will  intersect  at  large 
angles,  say  between  30°  and  120°. 

The  position  of  the  point  sought  in  relation  to 
the  three  located  points  from  which  position  of 
the  former  is  to  be  determined  may  be  (Fig.  1). 

1.  Inside  the  triangle  formed  by  the  three  lo- 
cated points. 

2.  Outside    the    triangle,    but    inside    the    great 
circle    (the   imaginary  circle  passing  through   the 
three  points  on  the  ground). 

•     3.  On  the  great  circle. 

4.  Outside  the  great  circle. 

In  all  cases  the  point  sought  is  on  the  same  side 
of  line  drawn  from  each  located  point,  that  is,  if 
on  the  right  side  of  one  line  looking  toward  the 
point  from  which  the  line  is  drawn,  it  is  on  the 
right  side  of  the  other  lines,  and  distant  from  the 
lines  drawn  from  the  three  points  in  proportion 
to  its  distance  from  each  point. 

12 


THREE-POINT  PROBLEM.  13 

When  the  plane  table  is  not  in  position  lines 
drawn  from  the  fixed  points  instead  of  intersecting 
at  one  point  will  form  a  triangle  called  the  triangle 
of  error,  unless  the  point  occupied  should  chance 
to  be  on  the  circle  passing  through  the  three  fixed 
points,  in  which  case  the  location  of  the  point 
occupied  is  indeterminate. 

1.   If  point  sought  is  within  the  triangle  on  the 


FIG.   1. 

ground,  in  other  words,  the  great  triangle,  the  true 
point  is  within  the  triangle  of  error  and  distant 
from  the  lines  from  the  three  located  points  in 
proportion  to  their  distances  from  the  point  occu- 
pied. 

2.  When  the  point  sought  lies  within  either  of 
the  three  segments  of  the  great  circle  formed  by 
the  sides  of  the  great  triangle,  the  true  point  is 
without  the  triangle  of  error  and  the  line  drawn 


14  PLANE  TABLE. 

from  the  middle  point  lies  between  the  true  point 
and  the  intersection  of  the  lines  from  the  other 
two  points. 

3.  When  the  point  sought  is  on,  or  very  near 
the  great  circle,  the  position  is  indeterminate. 

4a.  When  the  point  sought  is  without  the  great 
circle  and  the  middle  point  is  on  the  near  side  of 
the  line  joining  the  other  two  points,  as  is  the  case 
when  the  point  lies  inside  of  one  of  the  angles 
formed  by  the  sides  of  the  great  triangle  produced, 
the  true  point  is  without  the  triangle  of  error  and 
the  line  drawn  from  the  middle  point  lies  between 
the  true  point  and  the  intersection  of  the  other 
two  lines. 

46.  When  the  point  sought  is  without  the  great 
circle  and  the  middle  point  is  on  the  far  side  of 
the  line  joining  the  other  two  points,  the  true 
point  is  without  the  triangle  of  error,  and  on  the 
same  side  of  the  line  from  the  middle  point  as  the 
intersection  of  the  other  two  lines. 

The  following  rule  applies  to  all  points  outside 
of  the  great  circle:  The  point  sought  is  always 
on  the  same  side  of  the  line  from  the  most  distant 
point  as  the  intersection  of  the  other  two  lines  and 
distant  from  each  line  in  proportion  to  the  distance 
of  the  point  from  which  it  was  drawn.  In  case,  as 
occasionally  happens,  it  is  difficult  to  decide  which 
of  the  three  points  is  the  most  distant,  the  relation 
of  the  line  from  the  middle  point  to  the  lines  from 
the  other  two  points  as  described  in  4a  and  46  will 
determine  on  which  side  of  the  triangle  of  erupr 
the  point  sought  lies. 


/  OF  THE 

(    UNIVERSITY  ) 

A  OF  / 

X^L'Fo^NV^X 
THREE-POIN 


REE-POINT  PROBLEM.  15 

In  the  rare  case  where  the  three  located  points 
are  in  a  straight  line  the  same  rule  applies  as  in 
the  case  4a,  that  is,  the  line  from  the  middle  point 
lies  between  the  position  of  point  sought  and  the 
intersection  of  the  lines  from  the  other  two  points. 

The  above  rules  for  solving  the  three-point 
problem  form  what  is  commonly  called  the  method 
by  trial  and  is  the  one  usually  employed  in  the 
field  by  practical  plane-tablers. 

Other  methods  will  be  described  later 

In  making*  a  three  point  station,  if  the  lines 
drawn  from  the  three  fixed  points  form  a  triangle 
of  error,  showing  that  the  table  is  not  in  position, 
place  the  alidade  on  the  most  distant  point,  and 
over  the  point  on  the  paper,  inside  or  outside  of 
the  triangle  of  error,  as  the  case  may  be,  which  as 
near  as  can  be  judged  is  the  location  of  the  point 
sought.  Unclamp  the  table,  turn  it  in  azimuth 
until  the  far  point  is  sighted.  Clamp  the  table 
again,  and  draw  lines  from  the  three  fixed  points. 
If  these  lines  intersect  in  a  cohimon  point  the  table 
is  in  position.  If  not,  repeat  the  process,  until 
the  intersection  is  perfect.  In  short,  orient  on  the 
most  distant  point,  resect  on  the  nearer  points,  v. 

The  reason  for  orienting  on  the  most  distant 
point  is  that  any  movement  of  the  plane  table 
changes  this  point  more  in  azimuth  than  the 
nearer  points,  while  a  slight  turning  of  the  table 
would  have  but  little  effect  in  the  position  of  a 
nearby  point. 

However,  in  case  a  foresight  has  been  taken  to 


16  PLANE  TABLE. 

or  near  the  point  occupied,  it  is  well  to  orient  by 
this  foresight  and  resect  from  the  other  two  points. 
*  If  a  triangle  of  error  is  formed  it  will  usually  be 
small,  and  then  the  rules  applicable  to  the  three- 
point  problem  can  be  followed. 

After  the  three  point  station  is  made,  that  is, 
accurately  located,  the  plane- tabler  should  sight 
a  fourth  well  located  point  such  as  a  signal,  lone 
tree,  church  spire,  etc.,  if  any  are  visible,  as  a 
check  on  the  position,  and  if  a  line  from  this  point 
passes  through  the  station  occupied  making  a 
good  intersection,  (an  angle  between  30  and  120 
degrees)  the  station  may  be  considered  well  de- 
termined. Other  located  points  may  also  be 
sighted  as  a  test  of  the  accuracy  of  the  work. 

Sometimes  in  making  a  three-point  station  a 
mistake  in  position  of  one  or  more  of  the  located 
points,  from  which  the  station  is  to  be  determined, 
may  be  made;  that  is,  a  point  nearby  may  be 
mistaken  for  the  signal  at  the  located  point. 

In  this  case  it  is  well  to  remember  that  a  three- 
point  station  can  apparently  be  made,  that  is, 
lines  from  the  three  points  will  intersect  without 
forming  a  triangle  of  error,  although  the  location 
on  the  map  will  be  wrong,  and  if  the  true  point 
happens  to  be  very  near  "the  point  which  is  mis- 
taken for  it,  the  error  in  location  of  the  three- 
point  station  cannot  be  detected  unless  a  fourth 
located  point,  not  on  the  great  circle,  is  visible 
which  may  be  used  to  test  the  location  of  the 
station. 


THREE-POINT  PROBLEM.  17 

In  short,  the  fact  that  lines  supposed  to  be 
drawn  from  three  located  points  intersect  without 
a  triangle  of  error,  is  no  proof  of  correct  location  of 
station,  as  any  three  lines  will  come  in. 

Although  stations  located  in  the  manner  may  be 
but  slightly  out  of  position,  points  cut  in  from  such 
stations  may  be  greatly  in  error. 

However,  if  the  three  points  are  well  located 
and  there  is  no  doubt  about  their  identification,  a 
well  determined  three-point  station  can  be  made 
from  them,  •  and  afterwards  when  still  another 
point  has  been  located  from  the  three  initial  points 
the  accuracy  of  the  work  can  be  tested  as  although 
any  three  points  will  come  in,  lines  from  four 
points  can  not  meet  in  the  same  intersection,  unless 
they  are  correctly  located  in  reference  to  each 
other.  Although  they  may  appear  to  do  so,  if 
two  of  the  points  lie  in  nearly  the  same  direction 
from  the  station  occupied.  In  the  selection  of  a 
position  for  a  three-point  station,  care  should  be 
taken  that  the  point  chosen,  if  inside  the  triangle, 
is  not  nearly  in  line  between  two  of  the  three  points, 
or,  if  outside  the  triangle,  is  not  on  nor  very  near 
the  great  circle,  nor  so  far  outside,  that  lines  from 
the  three  points  intersect  at  such  angles,  less  than 
25  or  30  degrees,  that  the  intersection  is  indefinite, 
unless  a  fourth  point  is  discovered  that  will  serve 
as  a  check. 

Sometimes  the  plane-tabler  may  find  himself 
in  a  region  of  limited  view  where  only  short  sights 
can  be  obtained.  In  this  case,  when  orienting 


18  PLANE  TABLE. 

from  a  short  foresight,  great  care  must  be  taken 
to  sight  back  to  the  exact  point  from  which  the 
foresight  was  taken,  some  slender  object  as  a  pole 
having  been  left  there  to  indicate  the  precise  spot 
occupied,  and  the  points  that  are  used  for  resection 
should  also  be  small  and  sighted  with  extreme  care. 

If  great  care  is  not  exercised  in  this  respect,  on 
developing  the  triangulation  until  some  distant 
point  before  located  is  sighted,  it  will  probably 
fail  to  come  in,  that  is,  the  location  of  the  point 
occupied,  will  be  found  to  be  in  error,  although 
the  locations  made  by  the  short  sights  were  ap- 
parently correct  in  themselves. 

This  illustrates  the  extreme  hazard  of  carrying 
on  a  system  of  plane-table  triangulation,  based 
entirely  on  three  points,  and  emphasizes  the 
necessity  of  always  having  four  well  located  and 
distributed  points  in  view  from  every  station,  if 
possible. 

In  plane-table  work,  as  in  primary  triangulation 
a  system  of  quadrilaterals  is  to  be  preferred  to  that 
composed  of  plane  triangles.  In  making  three- 
point  stations  a  compass  will  be  found  a  conven- 
ience in  orienting  the  table  approximately. 

In  conjunction  with  the  stadia  or  telemeter,  the 
plane-table  may  be  used  in  locating  nearby  points 
by  radiation,  so  called. 

In  this  manner  with  one  or  more  rodmen  a  large 
number  of  points  can  often  be  located  from  one 
station. 

It  is  a  most  convenient  and  satisfactory  method 


THREE-POINT  PROBLEM.  19 

for  the  survey  of  lakes  and  rivers,  which  are  often 
so  situated  as  to  be  very  difficult  or  impossible  to 
locate  by  triangulation.  The  stadia-rod  can  also 
be  advantageously  used  in  a  survey  on  a  large 
scale,  where  it  is  desirable  to  locate  many  points 
within  a  small  area. 

To  make  a  stadia-rod,  take  a  strip  of  smooth 
unpainted  board,  from  four  to  six  inches  wide, 
and  ten  to  fifteen  feet  long.  Mark  off  the  board 
into  sections  six  inches  in  length  and  paint  each 
alternate  section  black. 

Set  up  the  plane  table  and  sight  the  rod  remov- 
ing it  to  such  distance  that  two  adjacent  stadia- 
hairs  will  cover  one  division  on  the  rod.  Measure 
distance  between  telescope  and  rod  carefully,  with 
tape  or  chain.  In  most  telescopes  the  stadia-hairs 
intercept  one  six-inch  division  on  the  rod  for  every 
hundred  feet.  If  this  should  not  prove  to  be  the 
case,  whatever  distance  is  intercepted  should  be 
taken  for  the  unit  of  distance.  For  more  exact 
work  stadia-rods  can  be  procured  of  makers  of 
engineering  instruments  graduated  to  any  re- 
quired fineness. 

To  locate  a  station  by  the  tracing  paper  method 
attach  the  paper  to  the  board  and  mark  a  point 
upon  the  paper  or  cloth  for  the  point  sought.  From 
this  point  sight  and  draw  lines  to  the  three  known 
points  from  which  the  station  is  to  be  determined. 
Then  shift  the  tracing  paper  until  each  of  the  three 
lines  passes  through  the  plotted  point  correspond- 
ing to  the  point  toward  which  it  was  drawn.  Posi- 


20  PLANE  TABLE. 

tion  of  the  point  sought  will  be  at  the  intersection 
of  these  lines,  which  can  be  pricked  through  the 
paper  on  to  the  plane-table  sheet.  This  method  is 
quite  often  used,  but  is  apt  to  be  inaccurate  unless 
great  pains  are  taken  in  drawing  the  lines  to  the 
sighted  points,  and  is  impracticable  in  windy 
weather. 

The  three-point  problem  may  be  solved  geo- 
metrically with  the  aid  of  compasses  as  follows: 

Let  a,  b,  and  c  be  the  projections  on  the  plane- 
table  sheet  of  the  three  located  stations  on  the 
ground  from  which  the  point  occupied  is  to  be 
determined. 

Draw  a  circle  through  a,  b  and  the  intersection 
of  the  lines  from  these  points  which  form  one  angle 
of  the  triangle  of  error;  also  draw  a  circle  through 
b,  c  and  the  corresponding  intersection.  A  third 
circle,  as  a  check  on  the  work  can  be  drawn  through 
a,  c  and  the  intersection  of  the  lines  from  these 
points. 

The  intersection  of  these  circles  is  the  location 
of  the  true  point. 

Place  the  alidade  on  this  point,  orient,  or  bring 
table  into  position,  by  sighting  one  of  the  fixed 
points  and  verify  the  position  by  resecting  on  the 
other  located  points.  This  method  is  inconven- 
ient in  the  field,  but  is  useful  in  the  office  as  an  aid 
to  the  proper  understanding  of  the  three-point 
problem. 

BesseVs  method  by  inscribed  quadrilateral,  or  the 
exact  method  so-called,  another  solution  of  the 


THREE-POINT  PROBLEM. 


21 


three-point  problem,  may  be  employed  when 
practicable,  which  is  not  always  the  case,  as  it 
often  happens  that  the  intersection'  of  the  construc- 
tion lines  comes  off  the  board. 


FIG.  2. 


In  Figs.  2  and  3,  a  b  c  are  points  on  the  plane- 
table  sheet  corresponding  to  the  located  points 
ABC  on  the  ground.  Set  up  the  table  at  the 
point  to  be  determined.  Place  the  alidade  on  the 
points  c  a  and  revolve  the  table  until  point  A  on 
the  ground  is  sighted.  Clamp  the  table,  and, 


22 


PLANE  TABLE. 


with  the  alidade  on  c,  sight  the  middle  point  B 
and  draw  a  line  c  e  along  the  edge  of  the  alidade 
rule.  Then,  with  the  alidade  on  the  line  a  c 
revolve  the  table  until  the  point  C  is  sighted. 

Clamp  the  table,  place  the  alidade  on  a,  sight 
the  middle  point  B  and  draw  the  line  a  e  along  the 
edge  of  the  rule.  A  line  drawn  through  e  and  the 
middle  point  b  will  pass  through  the  point  sought 


Set  the  alidade  on  this  line,  revolve  table  until 
point  B  is  sighted  and  table  will  be  in  position. 

Place  the  alidade  on  a  and  direct  to  A  and  draw 
a  line  along  the  alidade  rule. 

The  intersection  of  this  line  with  the  line  b  e  is 
location  of  point  sought. 

Verify  the  position  by  placing  the  alidade  on  c 
and  resecting  on  C. 


THREE-POINT  PROBLEM.  23 

The  following  analytic  solution  of  the  three- 
point  problem  is  adapted  from  Chambers'  Practical 
Mathematics. 

Given  the  distances  between  any  three  points 
and  the  angles  subtended  by  them  at  a  station  to 
find  the  relative  position  of  the  station  and  its 
distance  from  each  of  the  three  points. 

CASE  I. — When  the  station  is  outside  the  triangle 


formed  by  lines  joining  the  given  points  and  the 
middle  point  is  beyond  the  line  joining  the  other 
two  points  (Fig.  4). 

Let  A,  B,  C  be  the  three  points,  E  the  station 
occupied  and  m,  n  the  angles  read  from  E. 

Make  the  angle  A,  B,  D  =  m'  and  the  angle 
DAB  =  n'. 

In  the  triangle  ABC  the  three  sides  are  given 
hence  angle  A  can  be  found.  In  the  triangle 


24 


PLANE  TABLE. 


A  D  B,  the  angles  and  side  A  B  are  given;  hence 
A  D  can  be  found. 

In  triangle  A  D  C,  A  C  and  A  D  are  given,  and 
angle  A  =  C  A  B  —  DAB;  consequently  angle 
A  C  D  can  be  found. 

In  triangle  ACE  the  angles  and  side  A  C  are 
known;  therefore  the  sides  A  E  and  C  E  can  be 
found. 

Then  in  triangle  A  B  E  the  sides  A  B,  A  E  and 


the  angles  being  known  the  side  B  E  can  be  ob- 
tained. 

CASE  II. — When  the  station  occupied  is  without 
the  triangle,  and  the  middle  point  is  on  the  near 
side  of  the  line  joining  the  other  two  points  (Fig.  5). 

Let  the  middle  point  C  be  between  the  station  E 
and  the  line  A  B  then  the  point  E  and  D  will  be 
both  without  the  triangle  ABC  and  on  opposite 
sides  of  it,  and  the  solution  will  be  analogous  to 
that  of  the  first  case. 


THREE-POINT  PROBLEM.  25 

CASE  III. — When  the  station  is  within  the 
triangle  (Fig.  6). 

Let  D  be  the  station  then,  the  angles  A  D  C, 
B  D  C  being  given  their  supplements  are  known. 

Make  angles  A  B  E,  B  A  E,  respectively  = 
A  D  E  and  B  D  E. 

Angle  C  A  E  =  C  A  B +E  A  B. 

Angle  C  AD  =  180°  -(AC  D  +  AD  C}  and 

angle  BAD.  =  CAB-C  AD. 

In  the  triangle  A  C  D  the  angles  CAD  and 
ADC  are  known  and  the  side  C,  from  which  the 
sides  A  D  arid  C  D  can  be  found. 

In  the  triangle  BCD  the  angle  C  and  the  sides 
B  C  and  C  D  are  known  from  which  the  remaining 
side  B  D  can  be  found. 

*An  original  and  ingenious  graphic  solution  of  the 
three-point  problem  is  given  by  Prof. Llano  in  the  Engi- 
neering News  for  December  29,  1904. 


THE  TWO-POINT  PROBLEM. 

It  may  sometimes  be  desirable  to  place  the  table 
in  position  at  a  point  from  which  only  two  located 
inaccessible  points  can  be  seen. 

The  following  solution  of  this  problem  is  taken 
from  the  U.  S.  Coast  and  Geodetic-  Survey  Report 
for  1897-98:— Fig.  7. 

To  put  the  plane  table  in  position  at  a  third 
point  C  by  resection  from  two  located  points  A 
and  B,  whose  projections  on  the  sheet  are  repre- 
sented by  a  and  6,  select  a  fourth  point  D  so  that 
intersections  from  C  and  D  upon  A  and  B  will 
make  angles  sufficiently  large  for  good  determina- 
tions. 

Put  the  table  approximately  in  position  at  D, 
by  estimation  or  compass,  and  draw  lines  A  a, 
B  b,  intersecting  in  d.  Through  d  draw  a  line 
directed  to  C.  Then  set  up  at  C,  and  assuming 
the  point  c  on  the  line  d  C  at  an  estimated  distance 
from  d,  and  putting  the  table  in  a  position  parallel 
to  that  which  is  occupied  at  D,  by  means  of  the 
line  c  d,  draw  lines  from  c  to  A  and  from  c  to  B. 

These  will  intersect  the  lines  d  A ,  d  B  at  points 
a'  and  6',  which  form  with  c  and  d  a  quadrilateral 
similar  to  the  true  one,  but  erroneous  in  size  and 
position.  The  angles  which  a  b  and  a'  b'  make 
with  each  other  is  the  error  in  position. 

By  constructing  through  c  a  line  c  d'  making  the 
same  angle  with  c  d  as  that  which  a  b  makes  with 

26 


TWO-POINT  PROBLEM.  27 

a 'b',  and  directing  this  line  c  d'  to  77,  the  table 
will  be  brought  into  position  and  th  true  point  c 
can  be  found  by  the  intersections  of  a  A  and  b  B. 
Instead  of  constructing  with  drawing  instru- 
ments, the  angle  of  error  in  the  position  of  the 
table,  that  is  the  angle  of  the  line  a'  b'  makes 
with  the  line  a  6,  which  is  not  always  convenient, 
the  following  expedient  may  be  adopted. 


ar  b'  is  now  parallel  to  A  B  and  to  bring  the  table 
into  position  it  must  be  turned  until  a  b  is  parallel 
to  A  B. 

To  do  this  set  up  a  pole  or  other  mark  in  the 
direction  a'  b' \  set  alidade  on  a  b  and  revolve  the 
plane  table  until  a  b  points  to  the  mark. 

Then  a  b  is  parallel  to  A  B  and  the  table  is 
oriented. 


28  PLANE  TABLE. 

Another  solution  of  the  two-point  problem  is  as 
follows  (Fig.  8) : 

After  the  table  has  been  placed  in  position  by 
estimation,  resect  upon  A  and  B,  the  two  lines 
intersecting  at  c.  The  angle  a  b  c  is  the  angle  sub- 
tended by  A  B  at  C  and  the  position  of  the  point 
occupied  (C)  must  be  on  the  circle  passing  through 
a  b  c. 

Select  a  fourth  point  D  nearly  at  right  angles 


to  b  c.     Sight  the  signal  D  and  draw  line  c  d. 

Occupy  D,  place  the  alidade  on  the  line  c  d  and 
sight  C,  thereby  bringing  the  table  into  a  position 
parallel  to  its  position  when  at  C.  With  the 
alidade  on  d  observe  the  signal  at  B  and  draw 
the  line  d  e  intersecting  c  b.  c  e  is  the  distance  of 
C  from  B.  Lay  this  distance  off  from  b  in  the 
direction  of  c  as  a  chord  of  the  circle  drawn 
through  a  b  c. 


TWO-POINT  PROBLEM.  29 

The  intersection  of  the  chord  with  the  circle  at  / 
is  the  true  location  of  station  C. 

The  two  point  problem  can  also  be  solved  with 
tracing  paper  as  follows: 

The  two  located  points  A  and  B  are  plotted  on 
the  plane-table  sheet  as  a  and  b  (Fig.  7).  It  is 
desired  to  find  the  position  of  a  third  point  C. 
Occupy  a  fourth  point  D,  so  placed  that  lines  drawn 
from  the  other  points  will  intersect  at  sufficiently 
large  angles.  Orient  the  table  by  estimation  or 
with  the  compass. 

On  the  tracing  paper  which  is  fastened  to  the 
board,  indicate  a  point  to  represent  the  location 
of  D.  Draw  lines  from  this  point  towards  the  sta- 
tions A  B  and  C.  Lay  off  the  line  d  c  the  estimated 
distance  to  scale  between  d  and  c.  Occupy  the 
station  C  and  bring  the  table  into  a  position  parallel 
to  that  at  D  by  sighting  back  to  D  on  the  line  c  d. 

Draw  lines  from  c  to  A  and  B,  intersecting  the 
lines  drawn  to  the  same  points  from  D,  in  a'  and 
b'.  The  angle  which  the  lines  a  b  and  a'  b'  make 
with  each  other  is  the  error  in  position  of  the  table. 
Move  the  tracing  paper  until  the  line  a'  b'  is 
brought  over  the  line  a  6,  thereby  bringing  the 
table  into  position.  Sight  and  draw  a  line  from, 
in  other  words  resect  on  A  and  B.  The  intersec- 
tion of  these  lines  will  give  the  position  of  C  which 
can  be  pricked  through  with  a  needle  to  the  sheet. 

When  possible  to  get  in  line  with  two  located 
points  a  two-point  stati  n  may  be  made  in  the 
following  manner.  Set  the  plane  table  up  any- 


30  PLANE  TABLE. 

where  in  line  with,  and  bring  into  position  by  sight- 
ing on,  the  located  points.  Sight  the  point  where 
a  station  is  to  be  made,  which  should  be  at  some 
point  affording  angles  large  enough  for  good  inter- 
sections, drawing  the  line  at  the  most  convenient 
place  on  the  sheet. 

Occupy  the  point  to  which  the  line  is  drawn, 
and  bring  the  table  into  position  by  placing  the 
alidade  on  the  line  and  sighting  back  to  the  point 
just  left  from  which  the  line  was  drawn. 

The  table  will  now  be  in  orientation  and  the 
position  of  the  station  can  be  determined  by  resec- 
tion on  the  two  located  points.  Frequently  sta- 
tions can  be  made  by  ranging  in  between,  or 
lining  up,  that  is,  getting  in  line  outside  of  two 
located  points.  A  foresight  may  be  taken  over  a 
chimney,  or  a  tree  on  a  hillside,  and  a  station  made 
on  the  hilltop  back  of  the  same. 


CENTERING  THE  PLANE  TABLE  OVER  THE  STATION. 

In  regard  to  placing  the  plane  table  directly 
over  a  primary  triangulation  station,  or  other  lo- 
cated point  to  be  used  as  a  station,  it  is  well  to 
remember  that  exact  centering  of  the  table  is  only 
necessary  on  maps  of  very  large  scale. 

For  instance  on  scale  of  1  :  45,000,  75  feet  on 
the  ground  is  represented  by  1/50  of  an  inch  on 
the  map  and  as  1/150,  or  at  the  extreme  1/200, 
of  an  inch  is  about  the  limit  of  human  vision,  if 
the  plane-table  is  set  within  few  feet  of  the  exact 
point  there  can  be  no  appreciable  error. 

Plane- tablers  have  sometimes  taken  a  great 
deal  of  time  and  trouble  to  remove  and  afterwards 
replace  a  signal  in  order  to  set  directly  over  the 
station  when  the  scale  of. the  map  was  so  small  as 
to  render  such  care  entirely  unnecessary. 

In  regard  to  errors  arising  from  imperfect  level- 
ing of  table,  it  has  been  shown  by  Josiah  Pierce  in 
his  essay  on  the  "  Economic  Use  of  the  Plane 
table,"  that  errors  from  this  cause  amount  to 
practically  nothing  unless  the  inclination  of  the 
board  is  very  great. 

A  plane  table  may  be  15  degrees  out  of  level 
before  the  maximum  error  in  the  measurement  of  a 
horizontal  angle  will  amount  to  one  degree. 


31 


VERTICAL  ANGULATION. 

The  process  of  obtaining  altitudes  by  vertical 
or  dip  angles  is  peculiarly  adapted  to  the  plane 
table.  The  vertical  angle  between  the  station 
occupied  and  any  located  point,  of  which  it  is 
desired  to  ascertain  the  elevation,  is  read,  the 
distance  between  the  two  points  is  carefully  mea- 
sured on  the  map  and  from  this  data,  after  the 
correction  for  curvature  and  refraction  is  made,  the 
difference  in  height  of  the  two  stations  is  deter- 
mined. 

Differences  in  altitude  or  elevation  may  be  com- 
puted from  a  table  of  natural  tangents,  but,  as 
this  is  a  rather  tedious  operation  when  a  large 
number  of  angles  are  to  be  calculated,  vertical 
angle  tables  adapted  to  the  scale  of  the  map  and 
containing  corrections  for  curvature  and  refraction 
are  generally  used. 

Every  point  whose  elevation  it  is  desired  to 
ascertain  with  any  degree  of  precision  should  be 
observed  from  two  or  more  stations  whose  heights 
are  well  determined,  and  the  stations  from  which 
elevations  of  other  points  are  to  be  obtained  should 
be  determined  by  a  series  of  vertical  angles  from 
two  or  more  stations.  Reciprocal  angles  are  not 
checks  and  elevations  should  not  depend  upon  them 
alone. 

Any  datum  can  be  assumed,  but  if  it  is  desired 
to  connect  the  results  obtained  from  vertical  angles 

32 


VERTICAL  ANGULATION.  33 

with  any  special  datum,  as  sea-level,  a  line  of  levels 
can  be  run  from  some  convenient  bench  mark  to 
one  or  more  signals  or  other  exact  and  well  deter- 
mined point  on  the  map. 

To  obtain  the  best  results  with  vertical  angles 
exact  points  must  be  sighted,  as  for  instance  a 
cross-piece  attached  to  a  signal,  the  height  of 
which  above  the  ground  is  known. 

The  height  of  the  telescope  above  the  station 
occupied  must  also  be  measured.  This  will  be 
found  to  be  ordinarily  about  4.5  feet. 

The  correction  for  curvature  of  xthe  earth  is 
0.667  of  a  foot  for  the  first  mile,  increasing  as  the 
square  of  the  distance,  or  two- thirds  the  square  of 
the  distance  in  miles  equals  the  curvature  in  feet. 

The  usual  formula  for  computing  the  combined 
curvature  and  refraction  is  0.574  into  square  of 
distance  in  miles  equals  curvature  and  refraction 
in  feet.  In  making  allowance  for  refraction  it  is 
usually  reckoned  as  one-seventh  of  the  curvature, 
but  may  vary  greatly  from  this,  although  for 
distances  of  8  or  10  miles  the  variation  of  the  re- 
fraction would  not  be  the  cause  of  any  large  error 
that  might  occur. 

Large  errors  in  vertical  angles  at  these  distances 
are  caused  by  too  coarse  .or  imperfect  graduation 
of  the  vertical  arc,  imperfect  adjustment  of  alidade, 
difficulty  in  leveling  the  alidade  because  of  windy 
weather,  error  in  location  of  point,  or  mistakes 
in  reading  the  angles.  With  an  arc  reading  to 
One  minute,  and  by  estimation  to  one- half  minute, 


34  PLANE  TABLE. 

at  a  distance  of  5  miles,  angles  taken  to  exact 
points  as  signals,  lone  trees  or  houses  should  check 
out  within  5  feet  under  favorable  conditions  of 
weather.  Up  to  distances  of  seven  or  eight  miles, 
elevations  of  even  wooded  summits  should  be 
obtained  with  no  greater  error  than  half  a  20  ft. 
contour  interval,  say  ten  feet.  In  careful  vertical 
angle  work  it  is  w^ell  after  reading  an  angle  to  invert 
the  telescope  and  observe  again,  taking  the  mean 
of  the  observations  and  thereby  eliminating  the 
errors  of  adjustment. 

The  following  illustration  taken  from  the  U.  S. 
Coast  and  Geodetic  Survey  Report  for  1880  will 
best  show  how  this  is  done: 

Telescope  direct: 

Level  direct,  reading +0°     1 ' 

Level  reversed .  .  0' 


Mean +0°    0'  .5 

Angle  to  point +2°  17' 


Elevation  (difference) 2°  16'   .  5 

Telescope  inverted: 

Level  direct,  reading 0°    2' 

Level  reversed 1 ' 


Mean 0°     I'  .5 

Point..  .+2°  12' 


Elevation  (difference) 2°  13'  .  5 

Mean ,  2°  15' 


VERTICAL  ANGULATION.  35 

It  will  be  seen  that  the  level  was  one-half 
minute  out  of  adjustment,  the  horizontal  wire  one 
and  one-half  minutes,  and  that  revolving  the 
telescope  about  itself  changed  its  relation  to  the 
index  on  the  vernier  by  1'. 

The  mean  is  free  from  all  errors  of  adjustment. 

Before  reading  the  vertical  angles  the  adjustment 
of  the  striding-level  should  be  tested,  and  the 
index-error  of  the  vertical  arc,  if  any,  noted.  If 
the  zeros  on  the  arc  and  vernier  coincide  when  the 
telescope  is  level,  there  is  no  index-error.  If 
they  do  not,  the  reading  must  be  noted  and  the 
correction  applied  to  the  angles. 

Most  alidades  of  recent  make  have  an  adjust- 
able vernier  thus  eliminating  the  index-error. 

An  example  of  a  vertical  angle  computation 
follows : 

Elevation  of  station  above  sea  250  ft.  Distance 
of  summit,  whose  height  is  to  be  obtained,  1.5 
miles.  Vertical  angle +1°  50'.  1.5  miles  =  7920 
ft.  Tangent  of  1°  50'  =  0.03201.  7920X0.03201 
=  253.5  ft.  1.3  =  correction  for  curvature  and 
refraction  and  4.5  ft. -height  of  telescope  above 
ground  at  station. 

253.5+1.3  +  4.5  =  259.3ft. 

250  +  259.3  =  509.3  ft.  height  of  summit. 

Angles  of  elevation  are  called  +  angles,  of  de- 
pression, —  angles. 

Although,  as  before  stated,  refraction  is  usually 
estimated  as  one-seventh  of  the  curvature,  it 


36  PLANE  TABLE. 

varies  greatly  to  an  indefinite  amount,  increasing 
with  the  distance  between  the  stations  observed. 

The  best  time  for  measuring  vertical  angles  is 
between  9  a.m.  and  3  p.m.,  as  between  these  hours 
the  vertical  refraction  is  less  variable  than  earlier 
or  later  in  the  day. 

A  book  should  be  used  in  which  to  record  vertical 
angles  as  well  as  such  descriptions  of  the  foresights, 
lines,  and  locations  as  may  be  desirable  as  an  aid 
to  identification,  but  at  the  same  time  it  is  well  to 
put  everything  on  the  plane-table  sheet,  as  num- 
bers, graphic  sketches,  and  written  descriptions, 
that  will  tend  to  make  the  work  legible  to  others 
than  the  plane-tabler  in  case  of  loss  of  record  book, 
or  the  plane-tabling  were  to  be  completed  by 
another.  The  Roman  numerals  may  be  used  for 
numbering  stations,  the  Arabic  figures  for  fore- 
sights, locations,  etc.,  and  the  altitudes  should  be 
placed  on  the  sheet  in  red. 


SIGNALS. 

Unbleached  cotton  cloth  is  best  for  signal  flags. 
It  wears  better  and  is  bleached  by  exposure  to  the 
air  in  a  few  days.  Flags  should  not  be  over  five 
or  six  feet  in  length,  as  longer  flags  will  not  blow 
out  in  light  winds.  If  a  large  flag  is  desired,  put 
up  two  flags,  one  under  the  other. 

White  is  better  than  black  or  colored  flags. 

Color  can  not  be  distinguished  at  a  distance  of  a 
few  miles  and  a  black  flag  will  not  show  plainly, 
if  at  all,  against  a  dark  background,  while  a  white 
flag  appears  black  against  a  bright  sky. 

The  signal  pole  may  be  from  fifteen  to  forty 
feet  in  length,  according  to  circumstances,  but 
should  at  least  rise  high  enough  above  the  ground 
to  allow  the  flag  to  blow  out  freely,  and  may  be 
nailed  to  a  tree,  or  fence,  or  set  firmly  in  the 
ground,  2.5  to  3  feet,  by  tamping,  as  on  occupying 
the  point  with  the  plane  table  it  will  not  be  neces- 
sary to  remove  the  signal  unless  the  work  is  on  a 
very  large  scale  as  perfectly  accurate  results  can 
be  obtained  by  setting  the  table  by  the  side  of  and 
within  a  few  feet  of  the  pole.  On  ledges  or  rocky 
hills  props  may  be  necessary  to  support  the  signal 
pole.  These  should  be  eight  feet  or  more  in 
length  according  to  height  of  pole  and  at  least  three 
in  number,  four  are  preferable,  set  120  degrees 
apart  and  nailed  to  the  pole  at  different  heights,  a 
foot  apart  if  possible,  to  prevent  the  pole  "  kick- 

37 


38  PLANE  TABLE. 

ing  out  "  in  high  winds.  The  lower  ends  of  the 
props  should  be  driven  firmly  into  the  ground  or 
else  weighted  down  and  held  in  place  by  heavy 
stones. 


LAND  SURVEYS. 

The  plane  table  is  well  adapted  to  the  survey 
of  large  or  small  tracts  of  land,  as  farms,  parks, 
et  cetera,  on  any  scale  desired,  except  in  heavily 
wooded  areas,  and  in  many  ways  is  more  con- 
venient and  efficient  than  the  trasit  or  Jacob-staff, 
as  the  work  is  plotted  directly  upon  the  sheet  in 
the  field  at  time  of  survey,  thereby  saving  possible 
errors  of  recording  and  plotting. 

In  beginning  the  survey  of  a  tract  of  land  of 
which  a  map  or  plat  is  to  be  made,  two  intervisible 
points  as  nearly  as  possible  on  a  level  should  be 
selected  and  the  distance  between  them  measured 
by  chain  or  tape. 

This  distance  will  serve  as  a  base-line.  These 
points  should  be  plotted  on  the  map  or  plan  at 
such  distance  apart  as  is  required  by  the  scale. 
Signals  should  be  erected  at  each  end  of  the  base 
line  and  also  at  as  many  other  prominent  points 
as  is  desirable  to  insure  proper  control. 

Set  up  the  table  at  one  end  of  base  line,  place 
alidade  on  the  plotted  points  on  the  sheet  which 
represent  the  stations  at  end  of  base-line,  and  re- 
volve table  until  signal  at  other  end  of  base-line 
is  sighted.  The  table  is  now  in  position. 

Sight  and  draw  lines  along  the  edge  of  the 
alidade  rule  to  all  the  visible  signals  as-  well  as 
other  prominent  points  it  may  be  desired  to  locate 
as  trees,  fence  corners,  houses,  sheds,  etc, 

39 


40  PLANE  TABLE. 

Then  occupy  the  other  end  of  base  line  and 
after  bringing  the  table  into  the  same  position  as 
at  first  station  by  sighting  back  to  same,  draw 
lines  to  all  visible  points  to  which  lines  were  drawn 
from  first  station,  whose  positions  are  such  that 
the  intersections  will  form  sufficiently  large  angles 
for  good  determinations.  After  completing  work 
at  this  station,  occupy  a  third  point  to  which  a  line 
or  foresight  has  been  drawn  and  locate  same  by 
resection  as  it  is  called,  that  is,  by  sighting  the 
first  station  occupied  at  end  of  base  line. 

From  this  last  station  other  points  can  be  inter- 
sected and  lines  drawn  to  new  points  that  come 
into  view. 

In  this  manner  the  work  can  be  carried  on  until 
the  entire  area  is  surveyed. 

The  position  of  points  that  can  not  be  located 
by  intersection,  not  being  visible  from  any  two 
stations,  may  be  found  by  stadia,  tape,  or  chain 
measurements,  or  even  by  pacing  if  the  distance  is 
not  great.  At  every  station,  sight  all  previous 
stations  and  locations,  as  each  location  should  be 
sighted  from  at  least  three  stations  as  a  check 
upon  the  accuracy  of  the  work. 

Altitudes  can  be  determined  by  vertical  angles, 
if  the  alidade  has  a  vertical  arc;  and  the  configura- 
tion of  the  ground  shown  by  contour  lines  drawn 
at  any  desired  vertical  interval. 

In  surveys  on  a  large  scale  care  must  be  taken  to 
set  the  plane  table  over  the  exact  point  repre- 
senting the  station.  In  government  surveys  the 


LAND  SURVEYS.  41 

locations  of  primary  triangulation  points  are  usually 
indicated  by  a  copper  bolt,  metal  tablet,  or  iron 
or  stone  post;  but,  for  temporary  purposes,  a 
chisel  mark  on  a  rock,  or  a  wooden  pin,  is  sufficient. 

On  a  scale  of  100  feet  to  1  inch,  0.01  of  an  inch 
on  the  map  represents  1  foot  on  the  ground.  In 
such  cases  to  insure  accuracy  in  the  plane-table 
work,  it  is  necessary  to  place  the  point  on  the 
table,  corresponding  to  the  station  on  the  ground, 
very  nearly  over  the  pin  marking  the  station,  and, 
in  order  to  do  this  properly,  a  plumb-line  should 
be  used  and  the  table  carefully  plumbed  over  the 
station. ' 

On  maps  of  large  scale  the  signal  poles  should 
be  placed  directly  over  the  pin  marking  the  point 
on  ground  and  great  care  taken  in  sighting  and 
drawing  lines  to  signals  and  other  objects  so  that 
the  exact  point  may  be  intersected  from  each  sta- 
tion. 

The  larger  the  scale,  the  more  care  is  required 
in  this  respect. 

In  order  that  the  plane  table  may  be  placed 
directly  over  the  station,  signals  should  be  built 
so  that  they  can  be  easily  removed  and  replaced, 
or  else  the  signal  pole  placed  .high  enough  from 
the  ground  to  clear  the  head  of  the  operator. 

The  magnetic  meridian  should  be  drawn  on  the 
sheet  and  if  desired  the  true  meridian  can  be  ob- 
tained by  a  north  star  observation. 


PLANE-TABLE  TRAVERSE. 

-On  the  U.  S.  Geological  Survey  the  plane  table 
is  used  largely  as  a  traverse  instrument. 

That  is,  in  surveying  or  traversing  roads,  trails, 
'Streams,  and  valley  ways,  the  table  being  oriented 
by  compass,  and  distances  obtained  by  wheel 
revolutions,  stadia,  tape,  or  pacing. 

Used  in  this  way,  it  is  the  most  satisfactory 
and  expeditious  method  of  filling  in  the  details 
of  a  map. 

For  this  purpose  a  smaller  and  lighter  plane- 
table  is  used  with  a  sight  alidade  made  especially 
for  the  purpose. 

This  alidade  is  a  brass  rule  six  inches  in  length 
and  fitted  with  a  front  and  back  sight  instead  of  a 
telescope,  the  short  distances  measured  in  traverse 
rendering  a  magnifying  glass  unnecessary. 


42 


PROJECTIONS. 

A  map  projection  is  an  approximately  rectangu- 
lar diagram  on  which  are  plotted  the  parallels  of 
latitude  and  meridians  of  longitude  according  to 
the  scale  of  the  map.  Many  different  projections 
have  been  invented  and  employed  from  time  to 
time,  but  the  polyconic  is  considered  the  most 
satisfactory,  as  it  involves  less  distortion  of  the 
earth's  surface,  and  is  the  projection  now  used 
by  all  U.  S.  Government  Surveys. 

The  U.  S.  Coast  and  Geodetic  Survey  publishes 
tables  of  this  projection  which  can  be  used  for  any 
scale,  and 

Bulletin  No.  234  of  the  U.  S.  Geological  Survey 
contains  tables  adapted  to  several  different  scales 
with  directions  for  laying-off  or  plotting  the  pro- 
jection. 

The  paper  used  for  the  field  sheet  must  be  of 
the  very  best  quality  as  it  is  a  substance  very  sus- 
ceptible to  changes  of  weather,  nor  can  fine  delicate 
lines  be  drawn  on  poor  paper. 

Even  the  best  paper  of  ordinary  thickness  has 
been  found  to  be  subject  to  considerable  distortion 
from  expansion  and  contraction  caused  by  changes 
in  the  weather  and  to  obviate  this,  two  sheets  of 
heavy  paragon  paper,  so  arranged  that  the  grain 
of  the  sheets  is  at  right  angles,  are  pasted  together 
with  muslin  between. 

It  has  been  found  that  the  slight  changes  that 
43 


44  PLANE  TABLE. 

occur  in  paper  prepared  in  this  manner  are  dis- 
tributed uniformly  in  all  directions,  thus  reducing 
the  distortion  to  a  negligible  quantity. 

Celluloid  sheets  are  sometimes  used  in  very 
rainy  regions,  but  with  proper  care  double-mounted 
paragon  eggshell  paper  can  be  used  in  almost  any 
climate  with  satisfactory  results. 

Sheets  of  this  double  thick  paper  can  not  be 
rolled,  but  must  be  transported  in  flat  tin  or 
wooden  boxes,  or  while  in  use  in  the  field  perma- 
nently attached  to  the  plane-table  board,  both 
when  not  in  use  being  kept  in  a  wooden  or  sole- 
leather  case. 

Single  thick  paragon  paper,  properly  seasoned, 
can  be  used  for  plane-table  work  with  good  results, 
if  care  is  taken  to  protect  it  from  the  weather,  and 
the  field  work  does  not  take  a  great  length  of  time. 

Various  appliances  have  been  used,  for  fastening 
the  plane-table  sheet,  or  map,  to  the  board,  as 
clamps,  screws,  and  thumbtacks. 

Clamps  are  objectionable  as  they  take  up  con- 
siderable space  on  the  board,  thereby  interfering 
with  the  use  of  the  alidade,  are  liable  to  slip,  and 
do  not  hold  the  paper  firmly  in  place. 

Decidedly  the  most  satisfactory  contrivance 
for  this  purpose  is  a  brass,  flat-topped  screw  which 
is  screwed  by  hand  into  a  cylindrical  brass  screw, 
with  an  inside  thread,  set  permanently  into  the 
board,  flush  with  the  surface  of  the  table.  This 
arrangement  holds  the  paper  firmly  in  position  and 
is  in  every  way  satisfactory. 


CONCLUSION. 

Two  pencils  should  be  used  in  plane-table  work, 
one  with  a  chisel  edge  for  drawing  lines  along  the 
edge  of  the  alidade  rule,  and  the  other  with  a 
cone  shaped  point  for  drawing  symbols,  numbers, 
etc.,  on  the  map.  These  pencils  should  be  very 
hard,  6  or  7  H  at  least  and  of  the  best  quality. 

For  recording  vertical  angles  etc.,  a  softer  pencil 
may  be  used.  Sandpaper  pads  for  sharpening  the 
pencils  and  rubber  pencil  tips,  for  erasing  pencil 
marks  are  a  great  convenience. 

A  fine  needle  with  a  large  head  made  by  melting 
sealing  wax,  should  be  used  for  marking  with  a 
small  needle-hole  the  exact  intersection  of  the 
lines  determining  the  position  of  a  station  "or  other 
located  point. 

Before  beginning  the  plane-table  work,  it  is  a 
good  plan  to  ride  over  a  considerable  portion  of 
the  area  to  be  surveyed  in  order  to  get  a  general 
idea  of  the  "  lay  of  the  land,"  position  and 
distribution  of  the  commanding  points,  and  amount 
of  control  needed. 

During  this  reconnaissance  the  main  stations 
can  be  selected  and  signals  erected,  leaving  minor 
stations  to  be  chosen  as  occasion  arises  during  the 
prosecution  of  the  work. 

In  this  way  the  progress  of  the  work  can  be 
facilitated  and  an  estimate  made  of  the  time  re- 
quired to  complete  the  triangulation. 

45 


46  PLANE  TABLE. 

Sometimes  it  will  be  found  necessary  to  return 
to  a  station  a  second,  or  even  a  third  time,  because 
of  details  at  first  overlooked  or  observations  pre- 
vented by  bad  weather. 

The  amount  of  control,  that  is,  the  number  and 
distribution  of  the  stations,  intersections  and  other 
located  points,  necessar  to  insure  the  required 
accuracy  in  the  map,  varies  according  to  the 
character  of  the  land  surface  and  amount  of  detail, 
and  must  largely  be  left  to  the  judgment  of  the 
plane-tabler;  but,  in  general,  it  may  be  said,  if 
the  plane-table  locations  are  properly  distributed, 
and  in  sufficient  number  to  insure  that  all  details 
are  shown  without  appreciable  error  to  scale,  the 
map  is  practically  perfect. 

One  great  advantage  the  plane  table  possesses 
over  other  surveying  instruments  is,  that  the  map 
is  made  directly  in  the  field  and  at  every  station 
all  visible  points  previously  located  can  be  tested 
and  any  error  discovered  and  corrected  at  once,  and 
not  as  sometimes  happens  in  transit  work,  remain 
undetected  until  long  after  when  the  angle  is 
plotted  in  the  office,  where  it  is  often  impossible 
to  rectify  the  error. 

In  short  the  plane-table  method  is  the  most 
rapid,  economical  and  satisfactory  way  of  making 
a  map  yet  discovered.  It  is  accurate  to  scale, 
detects  and  corrects  error  during  the  continuance 
of  the  field  work  and  as  before  remarked  is  well- 
nigh  a  universal  surveying  instrument  for  all  the 
purposes  for  which  a  map  is  intended,  as  has  been 


CONCLUSION.  47 

thoroughly    proven    by    many    years    use    on  the 
Government    surveys    in    all    kinds    of    country. 

Mr.  Gannett,  Geographer,  U.  S.  Geological 
Survey,  in  his  manual  of  Topographic  Methods 
says,  "  For  making  a  map  the  plane  table  is  a 
universal  instrument.  It  is  applicable  to  all 
kinds  of  country  to  all  methods  of  work,  and  to 
all  scales.  For  making  a  map,  it  is  the  most  simple, 
direct  and  economic  instrument;  its  use  render 
possible  the  making  of  the  map  directly  from  the 
country  as  copy,  and  renders  unnecessary  the 
taking  of  elaborate  notes,  sketches,  photographs, 
etc.,  which  is  not  only  more  expensive,  but  pro- 
duces inferior  results." 

In  conclusion,  the  following  extract  from  Pierce 
on  the  "  Use  of  the  Plane  Table  "  well  expresses  its 
merits  as  a  surveying  instrument: 

"  Recent  results  and  improvements  show,  be- 
yond a  doubt,  that  a  far  greater  degree  of  precision 
can  be  obtained  from  simple  graphic  triangulation 
than  is  commonly  supposed,  amply  sufficient  for 
purposes  of  mapping,  and  with  a  degree  of  economy 
unequalled  by  other  systems. 

"  The  engineer,  or  student,  can  not  afford  to  lose 
sight  of  a  valuable  and  simple  instrument,  with 
which  he  can,  even  alone  and  unassisted,  construct 
a  topographical  map  of  any  scale,  with  a  degree  of 
precision  only  limited  by  the  scale,  and  in  the 
same  time  that  he  would  occupy  in  taking  observa- 
tions with  other  instruments  " 


INDEX. 

Alidade,  adjustments  of 5 

telescopic 4,6 

sight 42 

Angulation,  vertical 32 

Collination 5 

Control 46 

Curvature 33 

Declination 3 

Land  surveys 39 

Leveling  the  plane  table 9,31 

Map  projections • 43 

Paper 43 

Parallax 5 

Pencils 45 

Plane  table,  advantages  of 8,  46,  47 

centering  over  station 31 

forms  of 3 

traverse 42 

triangulation 8 

Radiation 18 

Record  book 36 

Reciprocal  angles 32 

Refraction 33 

Signals 37 

Stadia-rod 19 

Striding- level 5 

Telemeter 18 

Telescope,  magnifying  power  of 7 

Three-point  problem 12,  19 

analytical  solution 23 

Bcssel's  method 21 

Geometrical  solution 20 

tracing-paper  method 19 

Traverse,  plane  table 42 

Triangle  of  error 11 

Two-point  problem 26 

Vertical  angulation 32 

49 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  5O  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


•       /    :. 


16  1961 


2  1936 


Ftb  27  1938 


KtC'D  L.Q 


JULXSWI 


MAB    lg  1940 


T— ~ 


101943 


MAR  01  1942 


LD  21- 


YB   10999 


/ 

x  /^^^^7 

TA 
176739 


